Clear[v,v2,eqs]

v[z_]:=v[z]=Simplify[Sum[ζ[z][i]λ[z][[i]],{i,1,Length[λ[z]]}]];

v[1]

v[1]/.ζ[1][_]->0

t=Tensor20;

Clear[v2]
v2[z_]:=Simplify[Table[Sum[t[[a,b,c]]v[z][[a]]v[z][[b]],{a,1,20},{b,1,20}],{c,1,20}]]

Clear[eqs]
eqs[z_]:=eqs[z]=Collect[caps.v2[z],ζ[_][_],Together];

eqs[1]

eqs[1]/.ζ[1][_]->0

eqs[ω];

Length[eqs[1]]

(eqs[1]==0)/.{d1->3.7,d2->4.2}

Solve[(eqs[1]==0)/.{d1->0.37,d2->0.42},Table[ζ[1][i],{i,1,4}]]

Solve[(eqs[1]==0)/.{d1->3.7},{d2}~Join~Table[ζ[1][i],{i,1,4}]]

Solve[(eqs[ω]==0)/.{d1->3.7},{d2}~Join~Table[ζ[ω][i],{i,1,2}]]

Solve[(eqs[ω]==0),{d1,d2}~Join~Table[λ[ω][i],{i,1,2}]]

Solve[(eqs[ω]==0)/.{d1->3.7,d2->4.2},Table[ζ[ω][i],{i,1,2}]]

Solve[(eqs[(ω^2)]==0)/.{d1->3.7,d2->4.2},Table[ζ[(ω^2)][i],{i,1,2}]]

roots={};

Do[AppendTo[roots,FindRoot[Take[eqs,7]/.ζ[1]->1/.{d1->3.7,d2->4.2},Table[{ζ[i],RandomReal[]},{i,2,8}],WorkingPrecision->30]];,{200}]

Sort[roots[[All,1,2]]]

Outer[Max[Abs[Chop[#1-#2]]]&,roots[[All,All,2]],roots[[All,All,2]],1];

Position[%,0]

DeleteCases[%,{a_,a_}]

J/.%

Det[%]

J=Table[D[Take[eqs,7]/.ζ[1]->1/.{d1->3.7,d2->4.2},ζ[i]],{i,2,8}]

Table[Max[Abs[Drop[eqs,7]/.ζ[1]->1/.{d1->3.7,d2->4.2}/.roots[[k]]]],{k,1,Length[roots]}]

Min[%]

NSolve[(eqs==0)/.{d1->3.7,d2->42.},Table[ζ[i],{i,1,8}]]